Prove $n(A-B)=n(A)-n(A \cap B)$

Question

Prove that: $n(A-B)=n(A)-n(A \cap B)$

This is an example from my book in which first step is like this:$$n(A)=n(A-B)+n(A \cap B) $$

But how did they get it.

Answer 1

For every element $x \in A$ we have either $x \in A \cap B$ or $x \in A - B$, but never both. Thus, the number of elements in $A$ is equal to the number of elements in $A \cap B$ plus the number of elements in $A - B$, i.e. $n(A) = n(A \cap B) + n(A - B)$.

Answer 2

Hint: Note that $A$ is the disjoint union of $A-B$ and $A\cap B$. To see this, draw a Venn diagram.